is slowly growing if it is positive, strictly increases to infinity and for every positive con- stant c x→∞lim α(cx)/α(x

POLONICI MATHEMATICI 55 (1991)

On approximation of analytic functions and generalized orders

by Adam Janik (Krak´ow)

Abstract. A characterization of a generalized order of analytic functions of several complex variables by means of polynomial approximation and interpolation is established.

We say that a differentiable function α defined on [0, ∞) is slowly growing if it is positive, strictly increases to infinity and for every positive con- stant c

x→∞lim α(cx)/α(x) = 1 .

In the sequel α and β are two fixed slowly growing functions.

Let K be a compact set in C^N, N ≥ 1, such that the Siciak extremal function of K ([6])

ΦK(z) := sup{|p(z)|^{1/deg p}: p a polynomial, deg p ≥ 1 , kpk ≤ 1}, z ∈ C^N, is continuous, k k being the supremum norm on K. Given a function g analytic in

KR:= {z ∈ C^N : ΦK(z) < R}

for some R > 1, we put

M (r) := sup {|g(z)| : ΦK(z) = r} , 1 < r < R . The quantity

% := lim sup

r→R

α log⁺M (r)
β(R/(R − r))

is called the (α, β)-order of g in the sense of Sheremeta ([4], [3]). If α = β = log⁺ (suitably modified near 0) and K is a ball, we obtain the classical definition of the order of an analytic function.

The aim of this paper is to characterize the (α, β)-order of a function g analytic in KR by means of polynomial approximation and interpolation to

1991 Mathematics Subject Classification: 32A22, 41A25.

g on K. A characterization of a similar generalized order of entire functions was established in [2].

Given a function f defined and bounded on K, we put for n ∈ N E_n⁽¹⁾= E_n⁽¹⁾(f, K) := kf − tnk ,

E_n⁽²⁾= E_n⁽²⁾(f, K) := kf − lnk , E_n+1⁽³⁾ = E_n+1⁽³⁾ (f, K) := kln+1− tnk ,

where tn denotes the nth Chebyshev polynomial of the best approximation to f on K and ln denotes the nth Lagrange interpolation polynomial for f with nodes at extremal points of K ([5]).

Theorem. Let K be a balanced compact set in C^N such that ΦK is continuous. For positive x and c write

F (x, c) := β⁻¹(cα(x)) . Assume that for every positive c

lim sup

x→∞

d log F (x, c) d log x < 1 ,

α(x/F (x, c)) = (1 + o(x))α(x) as x → ∞ .

Then a function f defined and bounded on K is the restriction to K of a function g analytic in KRfor some R and of finite (α, β)-order % if and only if

% = lim sup

x→∞

α(n)

β(n/ log⁺(En^(j)Rⁿ))

, j = 1, 2, 3 (with the obvious conventions 1/0 = ∞ and 1/∞ = 0).

We begin by proving the following

Lemma. Let the assumptions of the Theorem hold and let (pn)_n∈N be a sequence of polynomials in C^N. Assume that

(i) deg pn≤ n, n ∈ N,

(ii) there exist n0∈ N, µ > 0 and R > 1 such that

log⁺(kpnkRⁿ) ≤ n/F (n, 1/µ) provided n ≥ n0. ThenP∞

n=0pn is an analytic function in KRand its (α, β)-order % does not exceed µ.

P r o o f. From (ii)

log⁺(kpnkrⁿ) ≤ n log(r/R) + n/F (n, 1/µ)

provided n ≥ n0 and 1 < r < R. By the methods of calculus we find that the maximum of the function

R+3 x → x log(r/R) + x/F (x, 1/µ)

is reached for x = xr, where xr is the solution of the equation x = α⁻¹



µβ 1 − d log F (x, 1/µ)/d log x log(R/r)



.

From the assumptions of the Theorem and the properties of α and β we obtain

xr = (1 + o(1))α⁻¹(µβ(R/(R − r))) as r → R . Thus for r sufficiently close to R

(1) log⁺(kpnkrⁿ) ≤ const. α⁻¹(µβ(R/(R − r))), n ∈ N . For every polynomial p we have ([6])

|p(z)| ≤ kpkΦ^{deg p}_K (z) , z ∈ C^N. So for every r ∈ (1, R) the series P∞

n=0pn is convergent in Kr, whence P∞

n=0pn is analytic in KR. Write

M^∗(r) := sup{kpnkrⁿ: n ∈ N}, r ≥ 0 ,

%^∗:= lim sup

r→R

α(log⁺M^∗(r)) β(R/(R − r)) . According to inequality (1) we have

log⁺M^∗(r) ≤ const. α⁻¹(µβ(R/(R − r)))

for r sufficiently close to R. This immediately yields %^∗ ≤ µ. Moreover (see [1], 2.3(1)),

log⁺M (r) ≤ log⁺M^∗(√

rR) − log(1 −p r/R) . Thus

α(log⁺M (r)) β

 R

R − r



≤ α(log⁺M^∗(√

rR) − log(1 −pr/R)) β

 R

R −√ rR

 ·

β

 R

R −√ rR



β

 R

R − r

 ,

which gives (after passing to the upper limit) % ≤ %^∗and consequently % ≤ µ.

P r o o f o f T h e o r e m. Let g be a function analytic in KR, of (α, β)- order %. Write

γj := lim sup

n→∞

α(n)

β(n/ log⁺(En^(j)Rⁿ)), j = 1, 2, 3 ;

here En^(j) stands for En^(j)(g, K). We claim that γj = %, j = 1, 2, 3. It is known (see e.g. [7]) that

(2) E_n⁽¹⁾≤ E_n⁽²⁾≤ (n_∗+ 2)E_n⁽¹⁾, n ≥ 0 , (3) E_n⁽³⁾≤ 2(n_∗+ 2)E_n−1⁽¹⁾ , n ≥ 1 ,

where n∗ := ^n+N_n
. Thus γ3 ≤ γ2 = γ1 and it suffices to prove that γ1≤ % ≤ γ₃.

We first prove γ1≤ %. By definition of the (α, β)-order we have for every µ > %

log⁺M (r) ≤ α⁻¹(µβ(R)/(R − r)) provided r is sufficiently close to R. By Lemma 3.4 of [1]

E_n⁽¹⁾≤ M (r)

(r − 1)rⁿ , 1 < r < R , so

log⁺(E_n⁽¹⁾Rⁿ) ≤ − log(r − 1) − n log(r/R) + α⁻¹(µβ(R/(R − r))) for every n ∈ N and for r sufficiently close to R. Substituting r = rn, where

rn := R[1 − 1/F (n/F (n, 1/µ), 1/µ)] , yields

log⁺(E_n⁽¹⁾Rⁿ) ≤ − log(rn− 1) − n log[1 − 1/F (n/F (n, 1/µ), 1/µ)]

+ n/F (n, 1/µ) .

On account of the assumptions and the properties of the logarithm we obtain log⁺(E⁽¹⁾_n Rⁿ) ≤ 4n/F (n, 1/µ)

for sufficiently large n. Hence, by the properties of slowly growing functions, for every ε > 0 and for sufficiently large n

α(n)

β(n/ log⁺(En⁽¹⁾Rⁿ))

≤ µ + ε .

Owing to the arbitrariness of ε > 0 and µ > % we get after passing to the upper limit γ1≤ %.

Next we claim % ≤ γ3. Suppose γ3< %. Then for every µ ∈ (γ3, %) α(n)

β(n/ log⁺(E⁽³⁾n Rⁿ))

≤ µ provided n is sufficiently large. Thus

log⁺(E_n⁽³⁾Rⁿ) ≤ n/F (n, 1/µ)

and by the Lemma % ≤ µ, which contradicts the assumption µ < %.

Now let f be a function defined and bounded on K. Put γj := lim sup

n→∞

α(n)

β(n/ log⁺(En^(j)Rⁿ)), j = 1, 2, 3 . We claim that if γk is finite for k = 1, 2 or 3, then

g := l0+

∞

X

n=0

(ln+1− l_n)

is the required analytic continuation of f to KR and its (α, β)-order % is γj, j = 1, 2, 3. Indeed, for every µ > γk

α(n)

β(n/ log⁺(En^(k)Rⁿ))

≤ µ provided n is sufficiently large. Hence

E_n^(k)Rⁿ≤ exp(n/F (n, 1/µ)) .

By (2), (3) and the Lemma, g is analytic in KR and its (α, β)-order % is finite. So by the first part of the proof % = γj, j = 1, 2, 3, as claimed.

References

[1] A. J a n i k, A characterization of the growth of analytic functions by means of poly- nomial approximation, Univ. Iagel. Acta Math. 24 (1984), 295–319.

[2] —, On approximation of entire functions and generalized orders, ibid., 321–326.

[3] O. P. J u n e j a and G. P. K a p o o r, Analytic Functions—Growth Aspects, Res. Notes in Math. 104, Pitman, Boston 1985.

[4] M. N. S h e r e m e t a, On the connection between the growth of a function analytic in a disc and moduli of the coefficients of its Taylor series, Visnik L’viv. Derzh. Univ.

Ser. Mekh.-Mat. 2 (1965), 101–110 (in Ukrainian).

[5] J. S i c i a k, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322–357.

[6] —, Extremal plurisubharmonic functions in C^N, in: Proc. First Finnish-Polish Sum- mer School in Complex Analysis at Podlesice ( L´od´z 1977), University of L´od´z, 115–

152.

[7] T. W i n i a r s k i, Application of approximation and interpolation methods to the ex- amination of entire functions of n complex variables, Ann. Polon. Math. 28 (1973), 97–121.

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